Place the eggs in a saucepan, cover with cold water by 1 inch, and bring to a gentle boil over medium-high heat. Reduce to a simmer and cook 9–10 minutes, until the whites are set and yolks are just firm. Cool under cold running water, peel, and quarter.
Make the vinaigrette: In a large bowl, mash the minced garlic with the kosher salt to a paste. Whisk in the red wine vinegar, then slowly whisk in the olive oil until emulsified. Whisk in the black pepper.
Prep the vegetables: Cut the tomatoes into wedges; slice the cucumbers into 0.25-inch half-moons; thinly slice the green bell pepper; thinly slice the radishes; and thinly slice the scallions. Add the tomatoes, cucumbers, bell pepper, radishes, scallions, and niçoise olives to the bowl with the vinaigrette.
Toss gently to coat the vegetables evenly, then taste and adjust seasoning if needed. Arrange the dressed vegetables on a large platter.
Flake the oil-packed tuna into large pieces and arrange over the vegetables. Lay the anchovy fillets over the salad and tuck in the quartered eggs. Tear the basil leaves and scatter over the top. Drizzle any remaining vinaigrette from the bowl over the salad and serve at room temperature.